Expt 9: The Physical Pendulum

Experiment #9

The Physical Pendulum

Goals:

  • Measure the earth’s gravitational field strength, $g$, using a physical pendulum.

References

  1. Resnick, Halliday and Krane, Physics, 5th Ed. (John Wiley & Sons, Inc., 2002).

Introduction

In this experiment, you will investigate the dependence of the period of oscillation of a physical pendulum on the distance between the point of suspension and the pendulum’s centre of mass and use this relationship to determine the gravitational field strength near the earth’s surface.

A rigid body with distributed mass able to pivot about a horizontal axis which does not coincide with its centre of gravity is known as a physical pendulum or a compound pendulum. The physical pendulum that you will use in this experiment is a one-metre-long bar of steel which may be supported at different points along its length, as shown in the figure.

 A physical pendulum. The centre of mass is located at $CM$ while the pendulum oscillates around the pivot point O
A physical pendulum. The centre of mass is located at $CM$ while the pendulum oscillates around the pivot point O

The period of the physical pendulum can be calculated from the equation of motion of the bar. If we denote the distance between the point of suspension, O, and the centre of mass $CM$ by $l$, the period of this pendulum is given by

\begin{equation}
T~=~2~\pi~\left(\frac{k^2+l^2}{g~l}\right)^\frac{1}{2}
\label{physicalperiod}
\end{equation}

where $k$ is the radius of gyration of the bar about an axis passing through the centre of mass.

The behaviour of a physical pendulum of length $l$ can be related to that of a simple pendulum of length $l^\prime$, where the period of a simple pendulum of length $l^\prime$ is given by

\begin{equation}
T~=~2~\pi~\left(\frac{l^\prime}{g}\right)^\frac{1}{2} \quad . \label{simpleperiod}
\end{equation}

By equating Eqs. \ref{physicalperiod} and \ref{simpleperiod} and solving for $l$, we can derive an equation for the values of $l$ for which the pendulum has the same period of a simple pendulum of length $l^\prime$:

\begin{equation}
l~=~\frac{l^\prime \pm \left({l^\prime}^2~-~4~k^2 \right)^\frac{1}{2}}{2} \quad . \label{length}
\end{equation}

As you can see, there are two values of $l$, $l_1$ and $l_2$, for which the period of the physical pendulum is the same as that of a given simple pendulum. There is also a value of $l$ for which the physical pendulum has a minimum period. The minimum period may be found by minimizing Eq. \ref{physicalperiod}

\begin{equation}
\frac{dT}{dl}~=~0 \quad .
\end{equation}

Solving this equation yields a minimum period

\begin{equation}
T_{min}~=~2~\pi~\left(\frac{2 k}{g}\right)^\frac{1}{2} \quad . \label{minperiod}
\end{equation}

Prelab Questions

  1. Derive Eq. \ref{physicalperiod}. Show that $~l_1~l_2~=~k^2$ for a fixed $T$.
  2. Write down an expression for the radius of gyration $k$ in terms of the dimensions of the bar.
    1. Does Eq. \ref{physicalperiod} apply when:
      1. A large amplitude is used?
      2. When damping is present, either due to friction at the pivot or due to air resistance?
    2. If Eq. \ref{physicalperiod} does not apply, would the value you find for $g$ be too high or two low?
  3. Should the presence of holes in the bar be considered when calculating the theoretical value of $k$?
  4. Will the increased weights used in the optional experiment alter the effects of damping?

Apparatus

  • pendulum bar and ball bearing mount
  • 2 extra masses
  • metre stick
  • calipres
  • photogate
  • stopwatch

Experiment

  1. Determine the period of the physical pendulum for various values of $l$. Describe your procedure in your lab notebook including the number of measurements, the amplitude of the oscillations and the number of times each measurement was repeated. Explore the effect of the amplitude of the oscillation on your measurements.
  2. Plot the results and fit to obtain experimental values of $k$ and $g$. Show the fit and residuals on your plot. Check that the relationship $l_1~l_2~=~k^2$ for a fixed $T$ holds for several values of $l_1$ and $l_2$. Show that the minimum value of $T$ satisfies Eq. \ref{minperiod}.
  3. Compare the measured value of $k$ to the value you calculate from the bar’s dimensions.
  4. Compare the measured value of $g$ to the known value.

2007-2 — Revised BJF
2014-4 — Revised NA

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